{
  "id": "math/0105125",
  "version": "v2",
  "published": "2001-05-15T21:31:08.000Z",
  "updated": "2018-07-09T19:29:42.000Z",
  "title": "Tiling spaces are Cantor set fiber bundles",
  "authors": [
    "Lorenzo Sadun",
    "R. F. Williams"
  ],
  "comment": "Updated 2018 to version published in 2003. 11 pages, including 4 embedded postscript figures",
  "journal": "Ergodic Theory and Dynamical Systems 23 (2003) 307-316",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a Z^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that 1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), 2) only a finite number of tile types are allowed, and 3) each tile type appears in only a finite number of orientations. The proof is constructive, and we illustrate it by constructing a `square' version of the Penrose tiling system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-05-15T21:31:08.000Z",
      "abstract": "We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a Z^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that 1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), 2) only a finite number of tile types are allowed, and 3) each tile type appears in only a finite number of orientations. The proof is constructive, and we illustrate it by constructing a ``square'' version of the Penrose tiling system.",
      "comment": "11 pages, including 4 embedded postscript figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-07-09T19:29:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58F13",
      "54F15"
    ],
    "keywords": [
      "cantor set fiber bundles",
      "tiling space",
      "finite number",
      "tile type",
      "fairly general spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......5125S"
    }
  }
}